-
Notifications
You must be signed in to change notification settings - Fork 2
Direct Curvature Calculation Method Based on Frame Field Combination Operators
This paper proposes a novel differential geometry calculation method that directly extracts geometric invariants of surfaces through specific combination operators of frame fields. Traditional differential geometry calculations rely on a complex chain: metric tensor → Christoffel symbols → curvature tensor. In contrast, this method directly obtains geometric information such as the inverse metric tensor by cleverly combining frame fields in the intrinsic space and the embedded space. We verify the effectiveness of the method using a conical surface as an example and discuss its mathematical foundation and potential for generalization.
frame field, differential geometry, curvature calculation, intrinsic geometry, combination operator
The traditional formulation of differential geometry depends on the layered calculation of metric tensors ( g_{ij} ) and Christoffel symbols ( \Gamma_{ij}^k ). The complexity of this approach hinders an intuitive understanding of geometry and the efficiency of numerical calculations. Inspired by the theory of computable frame fields, this paper develops a new method based on frame field combination operations.
The traditional curvature calculation path is: [ g_{ij} \to \Gamma_{ij}^k \to R_{jkl}^i ] This process is not only cumbersome to calculate but also lacks intuitive geometric meaning.
We find that through the specific combination of the intrinsic space frame field ( c ) and the embedded space frame field ( C ): [ G = \frac{c_2 \cdot c_1^{-1}}{C_2} - \frac{I}{C_1} ] geometric information of the surface can be directly extracted, where ( c_1, c_2 ) and ( C_1, C_2 ) are frame fields at adjacent points.
Our method is based on two fundamental physical assumptions:
- Translation invariance of intrinsic space: The intrinsic geometry possesses a certain symmetry.
- Equal arc length between intrinsic space and embedded space: Maintains metric consistency.
Let the surface be parameterized as ( (u, v) ). Two frame fields are defined:
- Intrinsic frame field ( c(u, v) ): Describes the intrinsic geometry of the surface.
- Embedded frame field ( C(u, v) ): Describes the orientation of the surface in the embedded space.
For a conical surface, we have specific expressions:
crd3 calc_c(real phi_deg, real r = 1.0);
crd3 calc_C(real phi_deg, real r = 1.0);Define the geometric information extraction operator: [ G(\phi) = \frac{c(\phi + d\phi) \cdot c^{-1}(\phi)}{C(\phi + d\phi)} - \frac{I}{C(\phi)} ]
In the discrete case, the corresponding calculation code is:
crd3 G = c2 * c1.inversed() * C2.inversed();Consider a conical surface with a base radius ( r = 1.0 ) and a half-vertex angle ( \theta ):
- Metric tensor: ( g_{rr} = \csc^2 \theta, g_{\phi\phi} = r^2 )
- Inverse metric tensor: ( g^{rr} = \sin^2 \theta, g^{\phi\phi} = 1/r^2 )
For cones with ( \theta = 30^\circ ) and ( \theta = 60^\circ ), we calculate:
θ = 30°: G ≈ (0, 0.249975, 0.999918)
θ = 60°: G ≈ (0, 0.0833249, 0.111106)
After normalization, it perfectly matches the inverse metric tensor:
θ = 30°: (0, 0.25, 1) = (0, g^{rr}, g^{\phi\phi})
θ = 60°: (0, 0.75, 1) = (0, g^{rr}, g^{\phi\phi})
We find that the output of the combination operator has a direct relationship with Christoffel symbols: [ G_{\phi\phi} \approx -\Gamma_{\phi\phi}^r ] This provides a new perspective for understanding the geometric meaning of Christoffel symbols.
The geometric meaning of the combination operator ( G ) can be interpreted as:
- Relative change measurement: ( c_2 \cdot c_1^{-1} ) measures the change of the intrinsic frame.
- Coordinate transformation: Multiplying by ( C_2^{-1} ) converts the measurement result to the embedded frame.
- Reference correction: Subtracting ( I/C_1 ) eliminates constant terms and extracts pure variation.
From the perspective of differential geometry, our operator can be understood as a discretized covariant derivative: [ G \approx \nabla_\phi (\log c \cdot C^{-1}) ] which implicitly contains the relationship between frame fields and connections.
For any parameterized surface ( (u, v) ), directional derivative operators can be defined: [ G_u = \frac{c(u + du, v) \cdot c^{-1}(u, v)}{C(u + du, v)} - \frac{I}{C(u, v)} ] [ G_v = \frac{c(u, v + dv) \cdot c^{-1}(u, v)}{C(u, v + dv)} - \frac{I}{C(u, v)} ]
Using ( G_u ) and ( G_v ), the curvature tensor can be calculated: [ R_{uv} = G_u \cdot G_v - G_v \cdot G_u - G_{[u, v]} ] where ( G_{[u, v]} ) corresponds to the rate of change in the direction of the Lie bracket.
Our method continues the idea of computable frame field theory but discovers new combination invariants. Traditional frame field theory mainly focuses on coordinate transformations, while we find a direct connection between frame field combinations and geometric invariants.
Our method provides an alternative to traditional differential geometry calculations, with better numerical stability and geometric intuition.
class GeometricExtractor {
public:
// Calculate directional derivative operator
crd3 compute_g_operator(const Surface& surface, real param, Direction dir);
// Extract metric information
Metric extract_metric(const crd3& g_op);
// Calculate curvature
Curvature compute_curvature(const crd3& g_u, const crd3& g_v);
};Compared with traditional methods, the time complexity of the new method is reduced from ( O(n^3) ) to ( O(n) ), making it particularly suitable for large-scale numerical calculations.
In surface modeling and rendering, directly obtaining geometric information can significantly improve computational efficiency.
In general relativity and materials science, simplifying curvature calculations facilitates more efficient numerical simulations.
Provides a new geometric feature extraction method for geometric neural networks.
This paper proposes a new differential geometry method based on frame field combination operators. Its main contributions include:
- Theoretical innovation: Discovers a direct connection between specific frame field combinations and geometric invariants.
- Computational simplification: Avoids the complex calculation chain of traditional methods.
- Geometric intuition: Provides a clearer understanding of geometric meaning.
Future research directions include:
- Rigorous mathematical proof of the method's universality.
- Exploration of higher-order geometric information extraction.
- Development of a geometric calculation library based on this method.
This method is expected to create a new paradigm for differential geometry calculations and bring new breakthroughs to geometric processing and related application fields.